NATALIA BRYKSINA, University of Manitoba, 240 Wallace Building, 125 Dysart
Road, Winnipeg MB, R3T 2N2On the number and stability of limit cycles in an SIR model
with saturated incidence
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We use the theory of Lyapunov coefficients to estimate the number, and
characterize the stability, of limit cycles associated with an SIR
model that employs saturated incidence function of the general form:
g(I) = kIh/(1+aIh), where k, a and h are
parameters. This study confirms that the model may have at most two
limit cycles when h=2. Furthermore, it is shown that for the case
h=3, the model may have a maximum of three limit cycles. The
stability of these limit cycles is characterized based on the signs of
the Lyapunov coefficients.

In this talk, we propose an efficient splitting domain decomposition
method (S-DDM) for solving parabolic equations and study a splitting
ELLAM for convection-dominated diffusion problems in high-dimensions.
We apply the S-DDM scheme by incorporating the upstream methods to
simulate the groundwater contaminant transport in porous media.

Herring tag data is available starting from 1936 in British Columbia.
We are looking at herring stray rates between five geographically
distinct populations defined by the Department of Fisheries and Oceans
(DFO). We will create a multi-state population model to estimate
yearly stray rates between populations. Based on deviances from these
estimates we will identify years where there have been noticeable
quantitative changes in herring stray behaviour. Hopefully these
changes can be associated with recorded environmental indicators.
Special consideration will be given to how to treat missing data
resulting from both fishery closures and missed tagging events.

In real application, it is usually difficult to obtain accurate IF
estimation of signals in terms of their multi-component feature and
the presence of noise. One kind of popular and useful approach for
measuring the IF is based on the property of time-frequency
representation. In the talk, we give a short review about two basic
techniques of them: peak detection method and the reassignment method.
And then we extend such technique with the recently developed
time-frequency representation: the Stockwell transform. The
improvement of our new methodologies are shown by numerical
simulations.

With the help of the Graffiti system, Fajtlowicz conjectured around
1992 that the average distance between two vertices of a connected
graph G is at most half the maximum order of an induced bipartite
subgraph of G, denoted a2(G). We prove a strenghtening of
this conjecture by showing that the average distance between two
vertices of a connected graph G is at most half the maximum order of
an induced forest, denoted F(G). Moreover, we characterize the
graphs maximizing the average distance among all graphs G having a
fixed number of vertices and a fixed value of F(G) or a2(G).
Finally, we conjecture that the average distance between two vertices
of a connected graph is at most half the maximum order of an induced
linear forest (where a linear forest is a union of chains).

This is joint work with Pierre Hansen (HEC Montréal) and Alain
Hertz, Rim Kilani and David Schindl (Ecole Polytechnique de
Montréal).

The Spekkens toy model is an interesting example of how to augment
classical physics in order to perform several quantum informational
tasks using limited resources. We revisit the Spekkens toy model and
look at the different representations for the group of operations on a
single toy bit. We show that in the representation of the operators
as Euler rotations, there exist rotations that obey the knowledge
balance principle, yet are not present in Spekkens' original group.
We demonstrate that this expanded group of single toy bit operations,
which includes Spekkens' original operations as a subgroup, is
isomorphic to the extended Clifford group for one qubit (modulo scalar
multiples of the identity). We also investigate the case for two toy
bits again expanding the group of toy operations to include some, but
not all, of the extended operations.

In this talk, we will show that the number M(n,k) of partitions of
nonnegative integer n with k parts can be described by a set of
[(k)\tilde] polynomials of order k-1 in Q[(k)\tilde],
where [(k)\tilde] denotes the least common multiple of
1,2,...,k and Q[(k)\tilde] is the quotient of n when
divided by [(k)\tilde]. In addition, the sets of the
[(k)\tilde] polynomials are obtained explicitly for k=3,4,5,
and 6.

We use Galois descent to construct central extensions of twisted
forms of split simple Lie algebras over rings. These types of
algebras arise naturally in the construction of Extended Affine
Lie Algebras. The construction also gives information about the
structure of the group of automorphisms of such algebras.

It was shown recently that the central path can be bent along the
simplex path of Klee-Minty cubes. This lead to tightening the
iteration complexity bound of central path following interior point
methods. Further, intriguing analogs between edge-paths and central
paths arise. We conjecture that the order of the largest total
curvature of the central path is the number of inequalities, and that
the average diameter of a bounded cell of an arrangement is less than
the dimension. We substantiate these conjectures and prove a
continuous analog of the d-step conjecture.

The John-Nirenberg inequality characterizes functions in the space
BMO in terms of the decay of the distribution function of their
oscillations over a cube [JN, 1961]. In joint work with Galia Dafni,
we prove a John-Nirenberg type inequality for functions in the space
Qa (Rn), which is a modified version of the
conjecture by Essén, Janson, Peng and Xiao [EJPX, 2000]. We
construct a function, as a counterexample, to show the necessity for
this modification.

Given a weighted complete graph GK(V,EK), we study a network
design problem to find an edge set EÍEK such that the
graph G(V,E) is connected. The power of a vertex u in G is the
maximum weight of the edges in E incident with it. Minimizing the
maximum vertex power is polynomial time solvable, while minimizing the
number of critical vertices with this minimized maximum vertex power
is NP-hard. For any fixed e > 0 we present a
(3/2+e)-approximation algorithm for the latter problem, and
show that this ratio is tight.